Archive of Applied Mechanics

, Volume 88, Issue 4, pp 503–516 | Cite as

A numerical study on the nonlinear behavior of corner supported flat and curved panels

  • Gaurav Watts
  • M. K. Singha
  • S. Pradyumna


The nonlinear behavior of corner supported plates and curved shell panels is investigated here using the first-order shear deformation theory based on Marguerre’s membrane strains for shallow shells and von Kármán’s nonlinearity. The nonlinear differential equations are transformed into a set of nonlinear algebraic equations by using the element-free Galerkin method. The moving kriging shape function with two different types of correlation formulae (Gaussian and quartic spline) is employed here. After studying the effectiveness of the method, a detailed parametric study is conducted to examine the effect of support size on the displacements and bending moments of corner supported rectangular plates. Thereafter, the numerical study is extended to the nonlinear bending and stability behaviors of corner supported shallow cylindrical and spherical shell panels.


Element-free Galerkin method Moving kriging shape function Support size Large deformation Shallow shells 


  1. 1.
    Timoshenko, S.P., Woinowsky-Krieger, S.: Theory of Plates and Shells. McGraw-Hill, NewYork (1959)zbMATHGoogle Scholar
  2. 2.
    Rajaiah, K., Rao, A.K.: Collocation solution for point-supported square plates. ASME J. Appl. Mech. 45(2), 424–425 (1978)CrossRefGoogle Scholar
  3. 3.
    Azarkhin, A.: Bending of thin plate with three-point support. ASCE J. Struct. Eng. 118(5), 1416–1419 (1992)CrossRefGoogle Scholar
  4. 4.
    Wang, C.M., Wang, Y.C., Reddy, J.N.: Problems and remedy for the Ritz method in determining stress resultants of corner supported rectangular plates. Comput. Struct. 80(2), 145–154 (2002)CrossRefGoogle Scholar
  5. 5.
    Lim, C.W., Yao, W.A., Cui, S.: Benchmark symplectic solutions for bending of corner-supported rectangular thin plates. IES J. Part A Civil Struct. Eng. 1(2), 106–115 (2008)CrossRefGoogle Scholar
  6. 6.
    Batista, M.: New analytical solution for bending problem of uniformly loaded rectangular plate supported on corner points. IES J. Part A Civil Struct. Eng. 3(2), 75–84 (2010)CrossRefGoogle Scholar
  7. 7.
    Li, R., Wang, B., Li, P.: Hamiltonian system-based benchmark bending solutions of rectangular thin plates with a corner point-supported. Int. J. Mech. Sci. 85, 212–218 (2014)CrossRefGoogle Scholar
  8. 8.
    Li, R., Wang, B., Li, G.: Benchmark bending solutions of rectangular thin plates point-supported at two adjacent corners. Appl. Math. Lett. 40, 53–58 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Sahoo, S., Chakravorty, D.: Static bending of point supported composite hypar shell roofs. J. Struct. Eng. 34(2), 169–176 (2007)Google Scholar
  10. 10.
    Das, H.S., Chakravorty, D.: A finite element application in the analysis and design of point-supported composite conoidal shell roofs: suggesting selection guidelines. J. Strain Anal. Eng. Des. 45(3), 165–177 (2010)CrossRefGoogle Scholar
  11. 11.
    Raju, I.S., Amba-Rao, C.L.: Free vibrations of a square plate symmetrically supported at four points on the diagonals. J. Sound Vib. 90(2), 291–297 (1983)CrossRefGoogle Scholar
  12. 12.
    Utjes, J.C., Sarmiento, G.S., Laura, P.A.A., Gelos, R.: Vibrations of thin elastic plates with point supports: a comparative study. Appl. Acoust. 19(1), 17–24 (1986)CrossRefGoogle Scholar
  13. 13.
    Schwarte, J.: Vibrations of corner point supported rhombic hypar-shells. J. Sound Vib. 175(1), 105–114 (1994)CrossRefzbMATHGoogle Scholar
  14. 14.
    Chakravorty, D., Bandyopadhyay, J.N., Sinha, P.K.: Finite element free vibration analysis of point supported laminated composite cylindrical shells. J. Sound Vib. 181(1), 43–52 (1995)CrossRefzbMATHGoogle Scholar
  15. 15.
    Chakravorty, D., Bandyopadhyay, J.N., Sinha, P.K.: Free vibration analysis of point-supported laminated composite doubly curved shells—a finite element approach. Comput. Struct. 54(2), 191–198 (1995)CrossRefzbMATHGoogle Scholar
  16. 16.
    Demir, C., Izmirli, S.B.: The effects of support size on the vibration of the point supported plate. Int. J. Phys. Sci. 6(8), 1920–1928 (2011)Google Scholar
  17. 17.
    Daripa, R., Singha, M.K.: Nonlinear vibration characteristics of point supported isotropic and symmetrically laminated plates. J. Aerosp. Sci. Technol. 62(2), 83 (2010)Google Scholar
  18. 18.
    Naghsh, A., Azhari, M.: Non-linear free vibration analysis of point supported laminated composite skew plates. Int. J. Non-Linear Mech. 76, 64–76 (2015)CrossRefGoogle Scholar
  19. 19.
    Li, S., Liu, W.K.: Meshfree and particle methods and their applications. Appl. Mech. Rev. 55(1), 1–34 (2002)CrossRefGoogle Scholar
  20. 20.
    Belytschko, T., Lu, Y.Y., Gu, L.: Element-free Galerkin methods. Int. J. Numer. Methods Eng. 37(2), 229–256 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gu, L.: Moving kriging interpolation and element-free Galerkin method. Int. J. Numer. Methods Eng. 56(1), 1–11 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Krysl, P., Belytschko, T.: Analysis of thin plates by the element-free Galerkin method. Comput. Mech. 17(1–2), 26–35 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Krysl, P., Belytschko, T.: Analysis of thin shells by the element-free Galerkin method. Int. J. Solids Struct. 33(20), 3057–3080 (1996)CrossRefzbMATHGoogle Scholar
  24. 24.
    Bui, T.Q., Nguyen, T.N., Nguyen-Dang, H.: A moving Kriging interpolation-based meshless method for numerical simulation of Kirchhoff plate problems. Int. J. Numer. Methods Eng. 77(10), 1371–1395 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hale, J.S., Baiz, P.M.: A locking-free meshfree method for the simulation of shear-deformable plates based on a mixed variational formulation. Comput. Methods Appl. Mech. Eng. 241–244, 311–322 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Bui, T.Q., Nguyen, M.N., Zhang, C.: Buckling analysis of Reissner–Mindlin plates subjected to in-plane edge loads using a shear-locking-free and meshfree method. Eng. Anal. Bound. Elem. 35(9), 1038–1053 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Watts, G., Singha, M.K., Pradyumna, S.: Nonlinear bending analysis of isotropic plates supported on Winkler foundation using element free Galerkin method. Int. J. Struct. Civil Eng. Res. 4(4), 301–307 (2015)Google Scholar
  28. 28.
    Watts, G., Pradyumna, S., Singha, M.K.: Nonlinear analysis of quadrilateral composite plates using moving kriging based element free Galerkin method. Compos. Struct. 159, 719–727 (2017)CrossRefGoogle Scholar
  29. 29.
    Kant, T., Kommineni, J.R.: C\(^{0}\) finite element geometrically non-linear analysis of fibre reinforced composite and sandwich laminates based on a higher-order theory. Comput. Struct. 45(3), 511–520 (1992)CrossRefzbMATHGoogle Scholar
  30. 30.
    Reddy, J.N.: Mechanics of laminated composite plates and shells: theory and analysis. CRC Press, Boca Raton (2004)zbMATHGoogle Scholar
  31. 31.
    Khdeir, A.A., Librescu, L., Frederick, D.: A shear deformable theory of laminated composite shallow shell-type panels and their response analysis II: static response. Acta Mech. 77(1–2), 1–12 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Reddy, J.N.: Exact solutions of moderately thick laminated shells. J. Eng. Mech. 110(5), 794–809 (1984)CrossRefGoogle Scholar
  33. 33.
    Palazotto A. N., Dennis S. T.: Nonlinear analysis of shell structures. American institute of aeronautics and astronautics, Washington (1992).
  34. 34.
    Kundu, C.K., Sinha, P.K.: Post buckling analysis of laminated composite shells. Compos. Struct. 78(3), 316–324 (2007)CrossRefGoogle Scholar
  35. 35.
    Surana, K.S.: Geometrically nonlinear formulation for the curved shell elements. Int. J. Numer. Methods Eng. 19(4), 581–615 (1983)CrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Applied MechanicsIndian Institute of Technology DelhiNew DelhiIndia

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